Using a Genetic Algorithm with a Mathematical Programming Solver to Optimize a Real Water Distribution System

Water 2018, 10, x FOR PEER REVIEW   

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Figure 3. Pressures for each node after adding new tanks (unfeasible solution). (a) New storage tanks 

added (b) Pressure distribution. 

Figure 4. Pressures for each node after adding new tanks and PRVs (feasible solution). (a) New PRVs 

added (b) Pressure distribution. 

3.4. Genetic Algorithm 

Genetic algorithms (GAs) were invented by Holland [30] and introduced by Goldberg [44] as 

evolutionary type algorithms. GAs are adaptive methods that have been widely used to solve search 

and optimization problems. These algorithms are based on the genetic process of living organisms. 

Over generations, populations evolve according to the principles of natural selection and survival of 

the fittest individual. This theory was proposed by Charles Darwin [31]. 

Representation of individuals. A solution for the FRM network is represented as an individual, 

consisting of several chromosomes. A chromosome is a set of genes. A gene represents a value of the 

solution. The individual, for the FRM network, is coded as a string of binary, real, and real or integer 

values. Figure 5 shows the graphical representation of an individual. It has three chromosomes 

(valve, pipe, and tank). The valve chromosome is a gene with two attributes: ‘valve setting’ and 

‘status’. 

The attribute ‘valve setting’ is defined by the valve closure percentage which is converted to a 

head loss coefficient. This attribute is set as a real value. The ‘status’ attribute has a binary value. It is 

used to indicate the existence of a valve in a pipe. Hence, if there were a valve in the pipe, the assigned 

value would be 1; otherwise, the assigned value would be 0. This value of ‘status’ attribute is assigned 

based on the information of the existing network. For the FRM individual, this value is 0 because it 

does not have any valve. 

For the pipe chromosome, the value of the gene is the diameter of the pipe and it is taken from 

a list of eight available commercial diameters, defined prior to the optimization run (12.7, 19.05, 25.4,